a rectangle whose length is 4 times of its width has the area of 44 cm2 . find the perimeter and the length of the diagonal of the rectangle . give your final answer in the radical form 

Let's denote the width of the rectangle as \( w \). According to the problem, the length \( l \) is 4 times the width, so:

\[
l = 4w
\]

The area \( A \) of the rectangle is given as 44 cm², and the area of a rectangle is calculated as:

\[
A = l \times w
\]

Substituting the values:

\[
44 = 4w \times w
\]

\[
44 = 4w^2
\]

To find \( w \), divide both sides by 4:

\[
w^2 = \frac{44}{4} = 11
\]

Now, take the square root of both sides:

\[
w = \sqrt{11} \text{ cm}
\]

Since \( l = 4w \):

\[
l = 4\sqrt{11} \text{ cm}
\]

### Perimeter Calculation

The perimeter \( P \) of a rectangle is calculated as:

\[
P = 2(l + w)
\]

Substituting the values of \( l \) and \( w \):

\[
P = 2(4\sqrt{11} + \sqrt{11}) = 2(5\sqrt{11}) = 10\sqrt{11} \text{ cm}
\]

### Diagonal Length Calculation

The length of the diagonal \( d \) of a rectangle can be found using the Pythagorean theorem:

\[
d = \sqrt{l^2 + w^2}
\]

Substituting the values of \( l \) and \( w \):

\[
d = \sqrt{(4\sqrt{11})^2 + (\sqrt{11})^2}
\]

\[
d = \sqrt{16 \times 11 + 11} = \sqrt{176 + 11} = \sqrt{187} \text{ cm}
\]

### Final Answers

- The perimeter of the rectangle is \( 10\sqrt{11} \) cm.
- The length of the diagonal of the rectangle is \( \sqrt{187} \) cm.

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. If the area of a square is 44 cm², what would be its side length and perimeter?
2. What is the area of a rectangle if its length and width are \( 2\sqrt{11} \) cm and \( \sqrt{11} \) cm, respectively?
3. How would the diagonal change if the width was doubled while keeping the length the same?
4. What is the length of the diagonal of a rectangle whose sides are 7 cm and 24 cm?
5. What is the perimeter of a rectangle whose length is twice its width and has an area of 72 cm²?

**Tip:** When solving geometric problems, always sketch a quick diagram to better visualize the relationships between the elements like length, width, and diagonal.

A rectangle whose length is 4 times its width has the area of 44 cm². Find the perimeter and the length of the diagonal of the rectangle. Give your final answer in radical form.

Learn how to calculate the perimeter and diagonal of a rectangle where the length is 4 times its width and the area is 44 cm². This problem involves geometric concepts, radical expressions, and the Pythagorean theorem. Suitable for students in Grades 8-10.

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